Singular boundary conditions for Sturm-Liouville operators via perturbation theory

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F23%3A00560217" target="_blank" >RIV/61389005:_____/23:00560217 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4153/S0008414X22000293" target="_blank" >https://doi.org/10.4153/S0008414X22000293</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4153/S0008414X22000293" target="_blank" >10.4153/S0008414X22000293</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Singular boundary conditions for Sturm-Liouville operators via perturbation theory

Popis výsledku v původním jazyce

We show that all self-adjoint extensions of semibounded Sturm-Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say d. epsilon {1, 2}. This characterization generalizes the well-known analog for semibounded Sturm-Liouville operators with with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written asnnA Theta= A0 + B Theta B*,nnwhere A Theta is a distinguished self-adjoint extension and T is a self-adjoint linear relation in Cd. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form-bounded with respect to A Theta, i.e., it belongs to H-1( A0), with possible 'infinite coupling'. A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure that the perturbation is well defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations Theta.nnThe merging of boundary triples with perturbation theory provides a more holistic view of the operator's matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.nnAs an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

Název v anglickém jazyce

Singular boundary conditions for Sturm-Liouville operators via perturbation theory

Popis výsledku anglicky

We show that all self-adjoint extensions of semibounded Sturm-Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say d. epsilon {1, 2}. This characterization generalizes the well-known analog for semibounded Sturm-Liouville operators with with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written asnnA Theta= A0 + B Theta B*,nnwhere A Theta is a distinguished self-adjoint extension and T is a self-adjoint linear relation in Cd. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form-bounded with respect to A Theta, i.e., it belongs to H-1( A0), with possible 'infinite coupling'. A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure that the perturbation is well defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations Theta.nnThe merging of boundary triples with perturbation theory provides a more holistic view of the operator's matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.nnAs an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Canadian Journal of Mathematics

ISSN

0008-414X

e-ISSN

1496-4279

Svazek periodika

75

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

37

Strana od-do

1110-1146

Kód UT WoS článku

000838450800001

EID výsledku v databázi Scopus

2-s2.0-85133433258